Integrand size = 17, antiderivative size = 186 \[ \int \frac {\sqrt [4]{1+x}}{\sqrt [4]{1-x}} \, dx=-(1-x)^{3/4} \sqrt [4]{1+x}+\frac {\arctan \left (1-\frac {\sqrt {2} \sqrt [4]{1-x}}{\sqrt [4]{1+x}}\right )}{\sqrt {2}}-\frac {\arctan \left (1+\frac {\sqrt {2} \sqrt [4]{1-x}}{\sqrt [4]{1+x}}\right )}{\sqrt {2}}-\frac {\log \left (1+\frac {\sqrt {1-x}}{\sqrt {1+x}}-\frac {\sqrt {2} \sqrt [4]{1-x}}{\sqrt [4]{1+x}}\right )}{2 \sqrt {2}}+\frac {\log \left (1+\frac {\sqrt {1-x}}{\sqrt {1+x}}+\frac {\sqrt {2} \sqrt [4]{1-x}}{\sqrt [4]{1+x}}\right )}{2 \sqrt {2}} \]
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Time = 0.08 (sec) , antiderivative size = 186, normalized size of antiderivative = 1.00, number of steps used = 12, number of rules used = 9, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.529, Rules used = {52, 65, 338, 303, 1176, 631, 210, 1179, 642} \[ \int \frac {\sqrt [4]{1+x}}{\sqrt [4]{1-x}} \, dx=\frac {\arctan \left (1-\frac {\sqrt {2} \sqrt [4]{1-x}}{\sqrt [4]{x+1}}\right )}{\sqrt {2}}-\frac {\arctan \left (\frac {\sqrt {2} \sqrt [4]{1-x}}{\sqrt [4]{x+1}}+1\right )}{\sqrt {2}}-(1-x)^{3/4} \sqrt [4]{x+1}-\frac {\log \left (\frac {\sqrt {1-x}}{\sqrt {x+1}}-\frac {\sqrt {2} \sqrt [4]{1-x}}{\sqrt [4]{x+1}}+1\right )}{2 \sqrt {2}}+\frac {\log \left (\frac {\sqrt {1-x}}{\sqrt {x+1}}+\frac {\sqrt {2} \sqrt [4]{1-x}}{\sqrt [4]{x+1}}+1\right )}{2 \sqrt {2}} \]
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Rule 52
Rule 65
Rule 210
Rule 303
Rule 338
Rule 631
Rule 642
Rule 1176
Rule 1179
Rubi steps \begin{align*} \text {integral}& = -(1-x)^{3/4} \sqrt [4]{1+x}+\frac {1}{2} \int \frac {1}{\sqrt [4]{1-x} (1+x)^{3/4}} \, dx \\ & = -(1-x)^{3/4} \sqrt [4]{1+x}-2 \text {Subst}\left (\int \frac {x^2}{\left (2-x^4\right )^{3/4}} \, dx,x,\sqrt [4]{1-x}\right ) \\ & = -(1-x)^{3/4} \sqrt [4]{1+x}-2 \text {Subst}\left (\int \frac {x^2}{1+x^4} \, dx,x,\frac {\sqrt [4]{1-x}}{\sqrt [4]{1+x}}\right ) \\ & = -(1-x)^{3/4} \sqrt [4]{1+x}+\text {Subst}\left (\int \frac {1-x^2}{1+x^4} \, dx,x,\frac {\sqrt [4]{1-x}}{\sqrt [4]{1+x}}\right )-\text {Subst}\left (\int \frac {1+x^2}{1+x^4} \, dx,x,\frac {\sqrt [4]{1-x}}{\sqrt [4]{1+x}}\right ) \\ & = -(1-x)^{3/4} \sqrt [4]{1+x}-\frac {1}{2} \text {Subst}\left (\int \frac {1}{1-\sqrt {2} x+x^2} \, dx,x,\frac {\sqrt [4]{1-x}}{\sqrt [4]{1+x}}\right )-\frac {1}{2} \text {Subst}\left (\int \frac {1}{1+\sqrt {2} x+x^2} \, dx,x,\frac {\sqrt [4]{1-x}}{\sqrt [4]{1+x}}\right )-\frac {\text {Subst}\left (\int \frac {\sqrt {2}+2 x}{-1-\sqrt {2} x-x^2} \, dx,x,\frac {\sqrt [4]{1-x}}{\sqrt [4]{1+x}}\right )}{2 \sqrt {2}}-\frac {\text {Subst}\left (\int \frac {\sqrt {2}-2 x}{-1+\sqrt {2} x-x^2} \, dx,x,\frac {\sqrt [4]{1-x}}{\sqrt [4]{1+x}}\right )}{2 \sqrt {2}} \\ & = -(1-x)^{3/4} \sqrt [4]{1+x}-\frac {\log \left (1+\frac {\sqrt {1-x}}{\sqrt {1+x}}-\frac {\sqrt {2} \sqrt [4]{1-x}}{\sqrt [4]{1+x}}\right )}{2 \sqrt {2}}+\frac {\log \left (1+\frac {\sqrt {1-x}}{\sqrt {1+x}}+\frac {\sqrt {2} \sqrt [4]{1-x}}{\sqrt [4]{1+x}}\right )}{2 \sqrt {2}}-\frac {\text {Subst}\left (\int \frac {1}{-1-x^2} \, dx,x,1-\frac {\sqrt {2} \sqrt [4]{1-x}}{\sqrt [4]{1+x}}\right )}{\sqrt {2}}+\frac {\text {Subst}\left (\int \frac {1}{-1-x^2} \, dx,x,1+\frac {\sqrt {2} \sqrt [4]{1-x}}{\sqrt [4]{1+x}}\right )}{\sqrt {2}} \\ & = -(1-x)^{3/4} \sqrt [4]{1+x}+\frac {\tan ^{-1}\left (1-\frac {\sqrt {2} \sqrt [4]{1-x}}{\sqrt [4]{1+x}}\right )}{\sqrt {2}}-\frac {\tan ^{-1}\left (1+\frac {\sqrt {2} \sqrt [4]{1-x}}{\sqrt [4]{1+x}}\right )}{\sqrt {2}}-\frac {\log \left (1+\frac {\sqrt {1-x}}{\sqrt {1+x}}-\frac {\sqrt {2} \sqrt [4]{1-x}}{\sqrt [4]{1+x}}\right )}{2 \sqrt {2}}+\frac {\log \left (1+\frac {\sqrt {1-x}}{\sqrt {1+x}}+\frac {\sqrt {2} \sqrt [4]{1-x}}{\sqrt [4]{1+x}}\right )}{2 \sqrt {2}} \\ \end{align*}
Time = 0.19 (sec) , antiderivative size = 107, normalized size of antiderivative = 0.58 \[ \int \frac {\sqrt [4]{1+x}}{\sqrt [4]{1-x}} \, dx=-(1-x)^{3/4} \sqrt [4]{1+x}+\frac {\arctan \left (\frac {\sqrt {2} \sqrt [4]{1-x^2}}{\sqrt {1-x}-\sqrt {1+x}}\right )}{\sqrt {2}}+\frac {\text {arctanh}\left (\frac {\sqrt {2} \sqrt [4]{1-x^2}}{\sqrt {1-x}+\sqrt {1+x}}\right )}{\sqrt {2}} \]
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Result contains higher order function than in optimal. Order 9 vs. order 3.
Time = 0.91 (sec) , antiderivative size = 443, normalized size of antiderivative = 2.38
method | result | size |
risch | \(\frac {\left (-1+x \right ) \left (1+x \right )^{\frac {1}{4}} \left (\left (1-x \right ) \left (1+x \right )^{3}\right )^{\frac {1}{4}}}{\left (-\left (-1+x \right ) \left (1+x \right )^{3}\right )^{\frac {1}{4}} \left (1-x \right )^{\frac {1}{4}}}+\frac {\left (-\frac {\operatorname {RootOf}\left (\textit {\_Z}^{4}+1\right )^{3} \ln \left (-\frac {\operatorname {RootOf}\left (\textit {\_Z}^{4}+1\right )^{3} \left (-x^{4}-2 x^{3}+2 x +1\right )^{\frac {1}{4}} x^{2}+2 \operatorname {RootOf}\left (\textit {\_Z}^{4}+1\right )^{3} \left (-x^{4}-2 x^{3}+2 x +1\right )^{\frac {1}{4}} x +\operatorname {RootOf}\left (\textit {\_Z}^{4}+1\right )^{2} \sqrt {-x^{4}-2 x^{3}+2 x +1}\, x +\operatorname {RootOf}\left (\textit {\_Z}^{4}+1\right )^{3} \left (-x^{4}-2 x^{3}+2 x +1\right )^{\frac {1}{4}}+\operatorname {RootOf}\left (\textit {\_Z}^{4}+1\right )^{2} \sqrt {-x^{4}-2 x^{3}+2 x +1}+\operatorname {RootOf}\left (\textit {\_Z}^{4}+1\right ) \left (-x^{4}-2 x^{3}+2 x +1\right )^{\frac {3}{4}}-x^{3}-2 x^{2}-x}{\left (1+x \right )^{2}}\right )}{2}+\frac {\operatorname {RootOf}\left (\textit {\_Z}^{4}+1\right ) \ln \left (\frac {\operatorname {RootOf}\left (\textit {\_Z}^{4}+1\right )^{3} \left (-x^{4}-2 x^{3}+2 x +1\right )^{\frac {3}{4}}+\operatorname {RootOf}\left (\textit {\_Z}^{4}+1\right )^{2} \sqrt {-x^{4}-2 x^{3}+2 x +1}\, x +\operatorname {RootOf}\left (\textit {\_Z}^{4}+1\right )^{2} \sqrt {-x^{4}-2 x^{3}+2 x +1}+\operatorname {RootOf}\left (\textit {\_Z}^{4}+1\right ) \left (-x^{4}-2 x^{3}+2 x +1\right )^{\frac {1}{4}} x^{2}+2 \operatorname {RootOf}\left (\textit {\_Z}^{4}+1\right ) \left (-x^{4}-2 x^{3}+2 x +1\right )^{\frac {1}{4}} x +x^{3}+\operatorname {RootOf}\left (\textit {\_Z}^{4}+1\right ) \left (-x^{4}-2 x^{3}+2 x +1\right )^{\frac {1}{4}}+2 x^{2}+x}{\left (1+x \right )^{2}}\right )}{2}\right ) \left (\left (1-x \right ) \left (1+x \right )^{3}\right )^{\frac {1}{4}}}{\left (1+x \right )^{\frac {3}{4}} \left (1-x \right )^{\frac {1}{4}}}\) | \(443\) |
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Result contains complex when optimal does not.
Time = 0.23 (sec) , antiderivative size = 159, normalized size of antiderivative = 0.85 \[ \int \frac {\sqrt [4]{1+x}}{\sqrt [4]{1-x}} \, dx=-\left (\frac {1}{4} i + \frac {1}{4}\right ) \, \sqrt {2} \log \left (\frac {\sqrt {2} {\left (\left (i + 1\right ) \, x - i - 1\right )} + 2 \, {\left (x + 1\right )}^{\frac {1}{4}} {\left (-x + 1\right )}^{\frac {3}{4}}}{x - 1}\right ) + \left (\frac {1}{4} i - \frac {1}{4}\right ) \, \sqrt {2} \log \left (\frac {\sqrt {2} {\left (-\left (i - 1\right ) \, x + i - 1\right )} + 2 \, {\left (x + 1\right )}^{\frac {1}{4}} {\left (-x + 1\right )}^{\frac {3}{4}}}{x - 1}\right ) - \left (\frac {1}{4} i - \frac {1}{4}\right ) \, \sqrt {2} \log \left (\frac {\sqrt {2} {\left (\left (i - 1\right ) \, x - i + 1\right )} + 2 \, {\left (x + 1\right )}^{\frac {1}{4}} {\left (-x + 1\right )}^{\frac {3}{4}}}{x - 1}\right ) + \left (\frac {1}{4} i + \frac {1}{4}\right ) \, \sqrt {2} \log \left (\frac {\sqrt {2} {\left (-\left (i + 1\right ) \, x + i + 1\right )} + 2 \, {\left (x + 1\right )}^{\frac {1}{4}} {\left (-x + 1\right )}^{\frac {3}{4}}}{x - 1}\right ) - {\left (x + 1\right )}^{\frac {1}{4}} {\left (-x + 1\right )}^{\frac {3}{4}} \]
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Result contains complex when optimal does not.
Time = 1.41 (sec) , antiderivative size = 41, normalized size of antiderivative = 0.22 \[ \int \frac {\sqrt [4]{1+x}}{\sqrt [4]{1-x}} \, dx=\frac {2^{\frac {3}{4}} \left (x + 1\right )^{\frac {5}{4}} \Gamma \left (\frac {5}{4}\right ) {{}_{2}F_{1}\left (\begin {matrix} \frac {1}{4}, \frac {5}{4} \\ \frac {9}{4} \end {matrix}\middle | {\frac {\left (x + 1\right ) e^{2 i \pi }}{2}} \right )}}{2 \Gamma \left (\frac {9}{4}\right )} \]
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\[ \int \frac {\sqrt [4]{1+x}}{\sqrt [4]{1-x}} \, dx=\int { \frac {{\left (x + 1\right )}^{\frac {1}{4}}}{{\left (-x + 1\right )}^{\frac {1}{4}}} \,d x } \]
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\[ \int \frac {\sqrt [4]{1+x}}{\sqrt [4]{1-x}} \, dx=\int { \frac {{\left (x + 1\right )}^{\frac {1}{4}}}{{\left (-x + 1\right )}^{\frac {1}{4}}} \,d x } \]
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Timed out. \[ \int \frac {\sqrt [4]{1+x}}{\sqrt [4]{1-x}} \, dx=\int \frac {{\left (x+1\right )}^{1/4}}{{\left (1-x\right )}^{1/4}} \,d x \]
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